What's the best method to turn a character left/right without a slipping feet?
When you see that the walk cycle doesn’t work anymore I guess you have no other solution than put a new action above your walk cycle in the NLA.


First I would make the armature differently: Create a root bone on the bottom, make it the parent of controllers, pole targets and of the spine base (current root). This way you’re free to move your spine up and down during the walk.
Prepare a walk cycle with no slipping, there are several methods, I guess it’s too long to explain here.
Don’t give your root the Follow Path constraint, give it to your armature object. After that, select the curve, go in the Graph Editor, unfold Evaluation Time, in the N panel remove the Generator modifier.
To set the speed of the walk along the curve, play with the curve's Evaluation Time value in Object Data > Path Animation, and, in the NLA, play with the action clip's Scale (NLA N panel > Action Clip > Playback Settings).
Once you’re good with the speed, look at the moment where you need to fix the slipping, for example where the curve turn is too narrow, put the time vertical bar at this frame.
Create a copy of your walk cycle in the Dopesheet. In the NLA, select the strip with the walk cycle, press ShiftA > Select the new action, it creates a new strip in a new track.
Select the new strip, press Tab, go in the Dopesheet, edit the action so that the feet don’t slip. In the 3D View You can copy paste the pose from the previous walk cycle to have a logical transition between this walk cycle and the new action.